A068776 a(0) = 1; for n > 0, a(n) is the smallest triangular number which is a (proper) multiple of a(n-1).

1, 3, 6, 36, 2016, 1493856, 579616128, 11013286048128, 1004811205553955491328, 1897191992473259952000123882626029056, 5012064437680248664058937311304485563631765718940918773832320000

Offset: 0

Author

Klaus Brockhaus, Mar 01 2002

Keywords

• nonn

Examples

			a(2) = 6, since 6 = 2*a(1) and 6 is a triangular number.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..13

Crossrefs

Cf. A000217, A068142.

Programs

Mathematica

pm1[{k_}] := {1, k-1}; pm1[lst_] := Module[{a, m, v}, a=lst[[1]]; m=Times@@Rest[lst]; v=pm1[Rest[lst]]; Union[ChineseRemainder[{1, #}, {a, m}]&/@v, ChineseRemainder[{-1, #}, {a, m}]&/@v]]; nexttri[1]=3; nexttri[n_] := Module[{s}, s=(pm1[Power@@#&/@FactorInteger[4n]]^2-1)/8; For[i=1, True, i++, If[s[[i]]>n, Return[s[[i]]]]]]; a[0]=1; a[n_] := a[n]=nexttri[a[n-1]]; (* First do <= 3, pm1[lst] is the list of numbers less than their product and == 1 or -1 (mod every element of lst). nexttri[n] is the smallest triangular number >n and divisible by n. *)

PARI

{a068776(m)=local(k,q,n); k=1; q=k*(k+1)/2; while(q0,k++; q=q+k))}

Python

from itertools import islice
from sympy import sqrt_mod_iter
def A068776_gen(): # generator of terms
    a = 8
    while True:
        yield a>>3
        b = a+1
        for d in sqrt_mod_iter(1,a):
            if d==1 or d**2-1 == a:
                d += a
            if d&1 and d < b:
                b = d
        a = b**2-1
A068776_list = list(islice(A068776_gen(),12)) # Chai Wah Wu, May 05 2024

Extensions

Edited by Dean Hickerson, Mar 09 2002